3. Features Direct Methods for Image Processing in HREM
of Solving Aperiodic Structures
Searching Algorithm for Finding
Modulation waves in 4D
Fourier Maps

4. far from the Scherzer defocus
Deconvolution of a Single EM
far from the Scherzer defocus
Original image
Image of the final model
Image after Fourier recycling
Averaged image
Deconvoluted image

5. Original EM from Prof. N. Uyeda
Symmetry
averaging
ED from Prof N. Uyeda
Original EM from Prof. N. Uyeda
Two-Step Image Processing
Image
Decon-
volution
Phase
extension
Fourier
recycling
Search for defocus
Partial
structure
model
Complete

6. Image Processing of Bi-2212 Deconvolution FT-1 Phase extension
EM image from Dr. S. Horiuchi
Space group: N [Bbmb]
a = 5.42, b = 5.44, c = 30.5Å;
q = 0.21b* + c*
Deconvolution
FT-1 c b Bi Sr Cu Ca
Oxygen in Cu-O layer
Phase
extension

7. Features Direct Methods for Image Processing in HREM
of Solving Aperiodic Structures
Searching Algorithm for Finding
Modulation waves in 4D
Fourier Maps

8. What’s a Modulated Structure ?
T = 0 (mod t) or MOD (T, t) = 0
Commensurate modulation Þ
superstructures
T ¹ 0 (mod t) or MOD (T, t) ¹ 0
Incommensurate modulation Þ
incommensurate structures T

9. Schematic diffraction pattern of an incommensurate modulated structureb* q

10. Conclusion In the reciprocal space:
The diffraction pattern of an incommen-surate modulated crystal is the projection of a 4- or higher-dimensional weighted lattice
In the direct space:
An incommensurate modulated structure is the “hypersection” of a 4- or higher-dimensional periodic structure cut with the 3-dimensional physical space

11. Representation of one-dimensionally modulated incommensurate structures
Lattice vectors in real- and reciprocal- space

12. Structure-factor formula
Modulated
atoms
situated at their
average positions

13. Modified Sayre Equations in multi-dimensional space

14. incommensurate modulated structures
Strategy of solving
incommensurate modulated structures
i) Derive phases of main reflections
using
ii) Derive phases of satellite reflections
using
iii) Calculate the multi-dimensional Fourier map
iv) Cut the resulting Fourier map with the
3-D ‘hyperplane’ (3-D physical space)
v) Parameters of the modulation functions are
measured directly on the multi-dimensional
Fourier map

15. DIMS:
direct methods for incommensurate structures

16. Modulated atoms in g - Na2CO3
Na O1,3

17. Bi-2223
superconductor
Incommensuratemodulation revealed by the direct method

18. (PbS)1.18TiS2 composite structure
4-dimensional average structure solved by the direct method

19. Features Direct Methods for Image Processing in HREM
of Solving Aperiodic Structures
Searching Algorithm for Finding
Modulation waves in 4D
Fourier Maps

20. MIMS: automatic search in 4D Fourier maps

21. MIMS: searching in 3-dimensional space

22. MIMS: searching in 4-dimensional space

23. MIMS: output structure model

24. 4-Dimensional Structure Refinement

25. Multislice Method for
conventional structures
and
aperiodic crystals

26. Bi-2201 Setting B=0 Setting B=0 & M=0 Setting M=0 Variation of
Using experimental thermal motion (B) & modulation (M) parameters
Bi-2201
Variation of
dynamical-diffraction
amplitudes with
sample thickness
Setting B=0
Setting B=0 & M=0
Setting M=0

27. calculated with dynamical-diffraction amplitudes
Potential Maps of Bi-2201
calculated with dynamical-diffraction amplitudes
~100Å
~200Å
~300Å

28. Fourier sections of the superconductor Bi-2212

29. 2D section in a 4D Fourier map

30. Contour mapping

31. Contrast Adjustment